On 6/30/2013 5:19 AM, apoorv wrote:> An ordinal, like any object of our discourse can be described by a string of> Symbols. Suppose we consider the set S of all ordinals that can be described> By a finite string of symbols. Now S must be an ordinal. Because if it were> Not so, then its members must not form an unbroken chain. so, there is an ordinal> X which is not in S , while the successor of X or some bigger ordinal is in S.> But if X is not describable by a finite string, then the successor of X also> Cannot be so, nor any bigger ordinal.> Now S, being an ordinal cannot be in itself.> So S, finitely described as ' The set of all ordinals that can be described> By a finite string of symbols' Cannot be ' a set describable by a finite string of> Symbols'.> The set S must then not exist. Then the Set S must be the set of all ordinals,> As that is the only set whose members form a chain, that does not exist.> Thus the set S = set of all ordinals.> Whence, all ordinals must be describable by a finite string of symbols.> But then, the set of all ordinals is countable.> From which, we get that there is some countable limit ordinal that does not> Exist.> So where is the flaw in the above reasoning ?>

Pretty good.

This is why one must postulate the existence ofa limit ordinal (different from the null-classwhich can be thought of as satisfying the definitiona limit ordinal in a trivial sense).

Limit ordinals are treated differently from successorsin transfinite recursion as well, are they not?

Here it the thing. At the end of the nineteenthcentury, there had been a re-evaluation of Kant andprogress in these subjects had been influenced bythe writings of Leibniz. Frege invokes Leibniz informulating his logic (two things are the same ifone can be substituted for the other without lossof truth) and, as far as I know, Cantor also makescertain references to Leibniz of a more metaphysicalnature (I am less clear on Cantor's works).

Here is one of Leibniz' statements:

"All existential propositions, though true, are not necessary, for they cannot be proved unless an infinity of propositions is used, i.e., unless an analysis is carried to infinity. That is, they can be proved only from the complete concept of an individual, which involves infinite existents. Thus, if I say, "Peter denies", understanding this of a certain time, then there is presupposed also the nature of that time, which also involves all that exists at that time. If I say "Peter denies" indefinitely, abstracting from time, then for this to be true -- whether he has denied, or is about to deny -- it must nevertheless be proved from the concept of Peter. But the concept of Peter is complete, and so involves infinite things; so one can never arrive at a perfect proof, but one always approaches it more and more, so that the difference is less than any given difference."

Leibniz

There are certain questions mathematicians generallydo not ask. One of them is "what is an object?"

So long as one does not ask what an object is, onecan count objects in succession.

I am certain a more skeptical reader of your reasoningwill help you to improve it. But, you are arguing overthe assumption of an axiom. This is a matter of understandingand not reasoning. That Cantor saw these things ina metaphysical sense is one thing. But, it is notnecessary for every mathematician to interpret thesethings metaphysically.

The directionality needed to support Leibniz' remarkis in the Peano axioms:

m+1=n+1 -> m=n

The identity of the object zero is decided by theinfinity of identity statements of its successors.